LAUSR

We consider a proper coloring of a plane graph such that no face is rainbow, where a face is rainbow if any two vertices on its boundary have distinct colors. Such a coloring is said to be proper anti-rainbow. A plane quadrangulation G is a plane graph in which all faces are bounded by a cycle of length 4. In this paper, we show that the number of colors in a proper anti-rainbow coloring of a plane quadrangulation G does not exceed $$3\alpha (G)/2$$ , where $$\alpha (G)$$ is the independence number of G. Moreover, if the minimum degree of G is 3 or if G is 3-connected, then this bound can be improved to $$5\alpha (G)/4$$ or $$7\alpha (G)/6 + 1/3$$ , respectively. All of these bounds are tight.